Question

Evaluate the integral \[ \int \frac{\sin^{-1} \sqrt{x} - \cos^{-1} \sqrt{x}}{\sqrt{x} \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right)} \, dx \]

• A. \(\dfrac{2}{\pi} \left[ \sin^{-1} \sqrt{x(2x - 1)} + \sqrt{x(1 - x)} \right] + x + C\)
• B. \(\dfrac{8}{\pi} \left( \sqrt{x} \sin^{-1} \sqrt{x} - \sqrt{x(1 - x)} \right) - 2\sqrt{x} + C\)
• C. \(\dfrac{2}{\pi} \left[ (2x - 1)\sin^{-1} \sqrt{x} - \sqrt{x(1 - x)} \right] - x + C\)
• D. \(\dfrac{2}{\pi} \left[ (2x - 1)\sin^{-1} \sqrt{x} - \sqrt{x(1 - x)} \right] + x + C\)

Hint:

Use inverse trigonometric identities like \(\cos^{-1} \theta = \frac{\pi}{2} - \sin^{-1} \theta\) to simplify integrals.

Answer 1

The Correct Option is B

We use the identity \(\cos^{-1} \theta = \frac{\pi}{2} - \sin^{-1} \theta\). Substituting in the integrand: \[ \frac{\sin^{-1} \sqrt{x} - \left(\frac{\pi}{2} - \sin^{-1} \sqrt{x}\right)}{\sqrt{x} \left( \sin^{-1} \sqrt{x} + \frac{\pi}{2} - \sin^{-1} \sqrt{x} \right)} = \frac{2\sin^{-1} \sqrt{x} - \frac{\pi}{2}}{\sqrt{x} \cdot \frac{\pi}{2}} \] Then integrate the simplified expression: \[ \int \left( \frac{4}{\pi} \cdot \frac{\sin^{-1} \sqrt{x}}{\sqrt{x}} - \frac{2}{\pi\sqrt{x}} \right) dx \] Using the standard result: \[ \int \frac{\sin^{-1} \sqrt{x}}{\sqrt{x}} dx = \frac{\pi}{4} \left( \frac{8}{\pi} \left( \sqrt{x} \sin^{-1} \sqrt{x} - \sqrt{x(1 - x)} \right) \right) \] and \[ \int \frac{1}{\sqrt{x}} dx = 2\sqrt{x} \] Putting all together, we get the answer as: \[ \frac{8}{\pi} \left( \sqrt{x} \sin^{-1} \sqrt{x} - \sqrt{x(1 - x)} \right) - 2\sqrt{x} + C \]